We thank the reviewers for the help in improving the paper. We have now
responded to the new comments as detailed below.

> 1) [request] P. 8, ll. 156--161,
> "It is then assumed that, ..." Now, I understand that you allow the
> multiplicities to be real numbers in your model. Then, it is not clear
> to me how you remove superdroplets from the system. In case \xi_i=\xi_j=1,
> do you allow them to become \xi_i=\xi_j=0.5 after coalescence? Or do
> you keep one of the superdroplets without changing the multiplicity
> \xi_i=1, and delete the other superdroplet? What if they are \xi_i=1.6 and
> \xi_i=1.2? Will they become xi_i=0.4 and \xi_i=1.2 after coalescence? Or
> Do you remove the superdroplet i? This is obviously important for your
> study, because you are discussing the impact of the "jumps" of lucky
> superdroplets. Please explain the deletion rule without any ambiguity.

No, we don't have \xi_i=\xi_j=0.5 in that case. Instead, we remove
the superdroplet with the smaller particles. However, if we had
\xi_i=\xi_j=1.1, then both would be 0.55. If any of these then collide,
they would be removed, because xi is less than unity. To clarify this
further we have now changed "less than one" to "one or less than one".

Regarding \xi_i=1.6 and \xi_i=1.2, yes, we do then get xi_i=0.4 and
\xi_i=1.2 after coalescence.

The jumps are not related to the value of xi, but just to the number of
superdroplets containing lucky droplets. This was already explained in
section 4.b.

> `2) [request] P. 22, l. 432, "..., and that the number of particles is
> approximately constant." This is what I already asked in my previous
> review comment (7). It is great that you performed the 1-D simulations
> with 2L, 8L, and 64L, and confirmed that the results are insensitive to
> this change. However, I still cannot understand why the number of real
> droplets is approximately constant in the original setup.
> For the 1-D simulation with vertical extent 1L, we have 255 background
> droplets and 1 lucky droplet in the domain. Then, at the time when the
> lucky droplet grows to 50um, the number of background droplets reduces
> from 255 to 132. It is almost halved!
> For 3-D, the setup is more confusing to me. Because the grid is 4x4x4,
> in the column where the lucky superdroplet is located, you have only 16
> real droplets (128/(4x4)=8 superdroplets) in it on average. This is not at
> all sufficient for the lucky droplet to grow to , because 16 is much less
> than 123x2 (x2 is for the two lucky droplets). Am I missing something?
> Please clarify this point.

We agree with the referee regarding our previous statement about the
number density of background droplets being nearly constant, and have
decided to investigate this problem as part of our new simulations where
we have now done a specific experiment where we investigate the effect
of removing a significant fraction of droplets during the growth to 50
microns. We see that the effect is very small; see the orange lines in
panels (a) and (b) of Fig.8. Instead of our previous phrase about the
number of background droplets being nearly constant, we have now added
a paragraph addressing this problem in connection with Figure 7.

We have now also removed the presentation of our 3-D results, although
we still explain that the variations in droplet number density influences
the distribution of P(T).

-----------------------------------------------------------------------------
Response to Reviewer 2

> 1. Throughout the manuscript: the phrase "superdroplet algorithm"
> is unclear. Do the authors mean the general approach in which a single
> computational particle represents a multitude of similar cloud droplets,
> or a specific way to calculate evolution of the droplet spectrum resulting
> from droplet collisions (as, for instance, in Unterstrasser et al. GMD
> 2017). This is never explained in the manuscript.

We mean the former, so we have now added the sentence "This is referred to
as ``superdroplet algorithm.''", putting superdroplet algorithm in quotes.

> 2. L.27-30. This sentence is unclear: How superdroplet algorithm
> incorporates "random distribution of superdroplets" (I assume random
> in space, correct)?  "Monte Carlo algorithm" for what? I assume
> for collisions. "Within the volume around one mesh point" – why
> is that relevant? Nature does not know about "mesh points".

We agree with the referee and have now removed the reference to
mesh points in this sentence. We have now modified it as follows:
"through the random spatial distribution of superdroplets and
through the explicit Monte Carlo collision algorithm involved."

> 3. L. 16, 61, and other places. What are "turbulent aerosols"?  

We have now changed it to "turbulent clouds".

> 4. Bottom of p. 8. If the superdroplet algorithm used in the code the
> authors use is different from Shima et al., then the algorithm needs
> to be explained in detail. The advantage of the Shima's algorithm is
> that it is linear in the number of particles because each superdroplet
> is allowed to collide only with a single randomly-selected other
> superdroplet (in one time step) rather than allowing collisions with
> all other superdroplets (like in the traditional bin microphysics).
> I think the authors argue that the N^2 scaling in the latter case (N is
> the number of superdroplets) is not important because only collisions
> between superdroplets in one grid volume are allowed and there are
> only a few superdroplets per grid volume. Is this correct? Perhaps the
> difference is that the Pencil code considers collisions between real
> particles, that is, superdroplets with multiplicity of one.

Our approach is what is said in Section 2a, but on top of this,
Shima et al used the permutation technique that we don't use.
To clarify this further, we have now added the following:
"However, this is not used in the Pencil Code. Instead, we allow each
superdroplet to collide with all other superdroplets within one grid
cell to ensure the statistical accuracy of the results. This leads to
a computational cost of O(ns^2(t)), which does not significantly
increase the computational cost because ns(t) is relatively small
for cloud-droplet collision simulations."

Another reviewer suggested that "...as sub-stepping is used in Shima et
al., so each droplet may interact with several others within a single
model time step"

In the penultimate paragraph of section 7.1 of Shima et al 2009, it says
that "SDM is using [ns/2] randomly generated, non-overlapping candidate
pairs, and allows multiple coalescence for each pair."

To clearly explain the algorithm of Shima et al 2009, we revised our texts
as follows:
"To reduce the computational cost and make it linear in the number of
superdroplets per mesh point, ns(t), Shima et al. (2009) supposed
that each superdroplet interacts with only one randomly-selected
superdroplet per time step rather than allowing collisions with all the
other superdroplets in a grid cell (Shima et al. (2009) still allows
multiple coalescence for randomly generated, non-overlapping candidate
pairs in sub-time step), which is what Shima et al. (2009) referred to
as random permutation technique."

> 5. L. 176 and in several other places. It is unclear what
> "one-dimensional" (1D) versus "three-dimensional" (3D) means. 1D
> is just a column, with random initial positions of superdroplets,
> correct? What is then 3D? Is there any air flow included? If not, what
> is a difference between 1D and 3D?

In 3-D, we have many columns, each with a different spatial distribution
of droplets. This broadens the distributions of P(T), and we quantify
by how much. However, we have now removed the explicit reference to 3-D
simulations and just explain the general problem associated with it. This
is because turbulence is not involved and performing 1000 3-D simulations
with at least 128*16 superdroplets is not feasible. The sentence in that
line has now been removed.

> 6. L. 209-214, see 5 above. This discussion is unclear. If a lucky
> superdroplet falls only in the vertical (i.e., in one column), how other
> columns affect the outcome? How superdroplets (lucky and standard) move
> in 1D and 3D? For 3D, (1) should include all 3 spatial dimensions to make
> it clear. Is the air turbulence included in the calculations? If it is,
> the computational domain is miniscule.

Similar to comment 5, 3-D columns invoke the fluctuations of the number
density between columns as we explained, "In 3-D, however, the number
density of the 10 um droplets beneath the lucky one is in general not
the same as the mean number density of the whole domain. This leads
to yet another element of randomness that we discuss in this study."

We have now added the following below Eq.(2). "Droplets are only subject
to gravity and no turbulent airflow is simulated."

> 7. L. 281-282: Please better explain the MFT. Perhaps a reference to
> a paper or textbook would be useful.

What we meant by MFT is that the actual collision times are just
replaced by the mean collision times that are given by t_k=lambda_k^{-1}.
To clarify this better, we have now written: "By comparison, if fluctuations
are ignored, the collision times that are given by t_k=lambda_k^{-1}.
This is what we refer to as mean-field theory (MFT)."

> 8. L. 290 and caption to Fig, 4. "Approach I" - this only becomes
> obvious later in the paper.

We thank the referee for having noticed it. We have now removed the
reference to approach I in this location and in the captions of
Figures 4 and 5.

> 9. Eq. 16, formulation of the collision efficiency is unclear. What
> is r_star?  Some explanation here is needed. Is that related to the Long
> kernel (Long JAS 1974)?

We do experiments where E is quadratic in r for radii above 30 micron,
for example, and constant below. We call this radius r_* and consider
different values between 10 and 40 micron. To clarify this better, we
have now written "To demonstrate this, we assume E_k ~ r_k^2 when r_k
exceeds a certain arbitrarily chosen value r_* between 10 and 40 um,
and E_k=const below r_*."

No, it is not related to the Long kernel.

> 10. Bottom of p. 17. I still do not have a clear picture of various
> approaches tried in this study. I is obvious. II: randomly distributed
> in space, correct? What does "solve for the collisions...explicitly"
> mean? With or without superdroplets (i.e., large multiplicity or
> multiplicity of 1)? III: explain the Monte Carlo algorithm. IV:
> section 2a only touches upon the way collisions between superdroplets
> are considered. Overall, should one consider an approach used in the
> traditional DNS of particle-laden suspensions, where the key is the
> collision detection algorithm, that is, considering collisions only when
> the computational particles are close enough? Perhaps comparing I to IV
> with such a situation would make the discussion clearer. I have to say
> that the Table 3 provides very little help.

We are here only talking about different approaches to solving the
LDM, and not about general computational techniques for particle-laden
suspensions. To help avoiding a wrong impression, we have now inserted
"to solving the LDM" in the relevant sentence.

Regarding approach II, we have now replaced our phrase "solve for the
collisions...explicitly" by "and then determine the distance to the
next droplet within a vertical cylinder of possible collision partners
to find the collision time".

Regarding the Monte Carlo method, we have now rewritten this more
explicitly: "A third approach is to use the mean collision rate to
compute the probability of a collision within a fixed time interval.
We then use a random number between zero and one (referred to as Monte
Carlo method) to decide whether ..."

We wish to clarify that these approaches are not meant to be used in DNS,
but we rather use them to explain that the superdroplet approach is just
a combination of approaches II and III. This helps to understand that
the effects of fluctuations in the LDM enter in two separate ways.
Table 3 lists the basics about the four approaches in a concise way;
it is not meant to replace the now improved explanations from the text.

The detailed explanations are given in sections 3.d and 3.e, and Section
3.c was meant to introduce the idea of talking about four different
approaches to the LDM, and we hope that our changes have now clarified
this. Comparison between approaches I and IV is shown in Fig. 7 and the
corresponding discussions were in the last paragraph of section 4.

> 11. L. 421. Are the concentrations considered here realistic? 300 per cc
> certainly is. 3,000 per cc with 10 micron droplets gives around 10 g/m3
> of cloud water (if my math is correct), high but not unrealistic. 30,000
> gives 100 g/m3 of cloud water, unrealistic for cloud physics.

We agree that 10n_0 and 100n_0 are not realistic. This is only
to test the numerical sensitivities of simulations to the initial
number density of cloud droplets.

> 12. Fig. 12: The solid line does not look like the average in the right
> panel. Or maybe the line is the same in all panels. Please explain.

We checked that at 50 micron, the average times are 1.955, 1.943, and
1.960, which are close to the MFT value of 1.968. To clarify this, we
have now shown the average in orange and write "The thick solid line
gives the average collision time and agrees with that of MFT (thick
black line) within about a percent." One should also remember that
the average is dominated by contributions from long times, which may
not have been appreciated.

> 13.  Section 4d, starting in L. 524. Please explain what 3D means, see
> 5 above. Specifically, what makes droplet number to fluctuate between
> columns. Just the initial condition? And does the superdroplet initial
> position change? Or maybe there is nonvanishing airflow in 3D simulations?

It is because of different spatial distribution of droplets in different
columns. In the penultimate paragraph of section 2b, we explained it as
"The superdroplet algorithm is usually applied to 3-D simulations. If there is
no horizontal mixing, one can consider 1-D simulations. Moreover, we are only
interested in the column in which the lucky droplet resides. In 3-D, however,
the number density of the 10um droplets beneath the lucky one is in
general not the same as the mean number density of the whole domain.
This leads to yet another element of randomness that we discuss in
this study: fluctuations of the number density between columns."

Turbulent airflow is not invoked. We have now added the following at
the end of the paragraph below Eq.(2): "Droplets are only subject to
gravity and no turbulent airflow is simulated."

-----------------------------------------------------------------------------
Responds to Reviewer 3

> What has still not been addressed from the points I had raised in
> previous rounds are:
> - the discussed "approaches" I, II, III and IV are still referred
> to (as early as page 14) before being defined (only on page 17);

We agree with the referee that "approach I" was used too early, as was
also noticed by referee I. We have now removed the reference to this
before introducing it.

> - the discussion/conclusions sections lack any mention of the fact
> that the super-droplet simulations described in literature are performed
> for multiplicities several orders of magnitude larger than these covered
> in the paper.

We did discuss this in the third paragraph of section 2.b. In addition,
we also performed simulations with xi=50 in Fig.A1(b), which is around
the same order as in other studies.

> To comply with the AMS Software preservation, stewardship, and reuse
> guidelines1, please provide precise information on the version of
> PencilCode used for the study and archive this particular version at a
> persistent location (e.g., zenodo).

As presented in the acknowledgement and data availability section, The
Pencil Code is publicly available at https://github.com/pencil-code.
The version used for this study is Version v2021.02.20 of Feb 20,
2021, with the DOI: 10.5281/zenodo.4553325. We uploaded the simulation
setup, simulation data, and scripts for post-processing on Zenodo at
"http://10.5281/zenodo.4742786".

> page 4, line 76: some research groups call it "multiplicity",
> others "weighting factor" - perhaps worth mentioning?

We have not yet found a suitable reference where a different expression
was used. The superdroplet algorithm in our study is consistent with
the one from Shima et al, in which the superdroplet algorithm was
first presented in the meteorology community. For consistency and the
readability, we use "multiplicity" instead of other terminologies.

> page 4, line 84: droplets --> droplet

Our sentence may have been badly phrased, but there are many background
droplets, so we have now written "The model describes one large droplet
of 12.6um radius settling through a dilute suspension of background
droplets with 10 um radius. We hope that the current formulation makes
it clear that we referred here to the background droplets, which all
have the same radius of 10um."

> page 4, line 85: droplets --> droplet

We hope that the new formulation is now clearer.

> page 4, line 89: in K&S 2005, a bi-disperse size distribution is used,
> not a Poisson one, right?

No, K&S 2005 assumed a Poisson droplet size distribution. 

> page 4, line 92: in D&P 2017, there was also comparison with LDM,
> please be more specific to support "unlike"

We have now spelled out the specific difference
"we compare here with the distribution of cumulative
collision times, which is the key diagnostics of the LDM."

> page 5, line 98: what is a collision velocity

We have now replaced "collision velocity" by
"velocities of colliding droplets"

> page 5, line 104: first mention of dimensionality? isn't the preceding
> discussion also relating to 3D? what is a 3D version of LDM?

As we explained in paragraph 4 of Section 2.b, different vertical columns
are different from each other. This is ignored in the standard LDM.
We have now removed this statement.

> page 5, table 5: some rows start with capital letter, other no

We have now changed the upper case to the low case.

> page 6, figure 1: explain what (a), (b) and (c) refer to in the caption

We have now explained the caption by writing  "... with (a): xi_i > xi_j,
(b): xi_i < xi_j, and (c): xi_i = xi_j ..."

> page 6, line 116: not all mentioned models use the same formulation for
> dx/dt - please clarify that it is part of "local" model formulation

The definition of dx/dt is in the section describing the superdroplet
algorithm, which is later referred to as approach IV, but all the other
approaches model the same physics and the same Equations (1) and (2) are
used. Regarding the sentence in line 116 about the hydrodynamic force,
we have now moved it to just after Eq.(1) and write "and the hydrodynamic
force is modeled using Stokes law, so that"

> page 6, line 128: "we limit" - please mention how it is handled in
> Shima et al. 2009 as the paragraph in a way suggests it is the same,
> but it is not.

Our time step criterion is indeed similar to that of Shima+09 in that
the time step times the probability should be much smaller than unity,
so we have now referred to their paper. There are also differences
related to the random permutation technique, but this is relate to the
probability and not the time step as such.

Eq. (25) of Shima et al. 2009 is very similar to our Eqs. (3) and (4).
But Eq. (25) of Shima et al. 2009 is not used in Shima et al. (2009)
and is proposed as a future work.

> page 6, line 132: "background droplets" - this is LDM specific,
> please clarify the text so that a reader is not confused what refers
> to Shima et al., to presented formulation, and to LDM

It is quite obvious from equation (4) that superdroplets with
the same velocity do not collide with each other. Therefore, we have
now omitted this sentence.

> page 6, line 135: which "earlier work"?

Assuming E_ij=1 is a simple assumption we have made, so we have now
written "For the purpose of the present study, it suffices to limit
ourselves to the simplest, albeit unrealistic assumption of $E_{ij}=1$,
but we also consider in one case a slightly more realistic quadratic
dependence on the radius of the larger droplet."

> page 8, line 165: this is not precise (not true) as sub-stepping is
> used in Shima et al. so each droplet may interact with several others
> within a single model time step

Another reviewer suggested that "The advantage of the Shima et al
algorithm is that it is linear in the number of particles because each
superdroplet is allowed to collide only with a single randomly-selected
other superdroplet (in one time step) rather than allowing collisions
with all other superdroplets (like in the traditional bin microphysics)."

In the penultimate paragraph of section 7.1 of Shima et al 2009, it says
that "SDM is using [ns/2] randomly generated, non-overlapping candidate
pairs, and allows multiple coalescence for each pair."

To clearly explain the algorithm of Shima et al 2009, we revised our texts
as follows:
"To reduce the computational cost and make it linear in the number of
superdroplets per mesh point, ns(t), Shima et al. (2009) supposed
that each superdroplet interacts with only one randomly-selected
superdroplet per time step rather than allowing collisions with all
the other superdroplets in a grid cell (Shima et al. (2009) still allows
multiple coalescence for randomly generated, non-overlapping candidate
pairs in sub-time step), which is what Shima et al. (2009) referred to
as random permutation technique."

> page 8, line 166: "linear sampling technique" is not mentioned in
> Shima et al. paper.

We agree that this expression was not used by Shima et al 2009, so we have now
removed that part.

> page 8, line 170: "linear in the total number of superdroplets":
> this is very misleading, if not incorrect; if focusing on this aspect,
> please give proper quantitative estimation for such statements

It is indeed not correct, we've now corrected and explained
the scaling in the last paragraph of section 2.a as the following,
"To reduce the computational cost and make it linear in the number of
superdroplets per mesh point, ns(t), Shima et al. (2009) supposed
that each superdroplet interacts with only one randomly-selected
superdroplet per time step rather than allowing collisions with all the
other superdroplets in a grid cell (Shima et al. (2009) still allows
multiple coalescence for randomly generated, non-overlapping candidate
pairs in sub-time step), which is what Shima et al. (2009) referred to as
random permutation technique. This technique was also adopted by Dziekan
and Pawlowska (2017) and Unterstrasser et al.  (2020). However, this
is not used in the Pencil Code. Instead, we allow each superdroplet to
collide with all other superdroplets within one grid cell to maximize the
statistical accuracy of the results. This leads to a computational cost of
O(ns^2(t)), which does not significantly increase the computational
cost because ns(t) is relatively small for cloud-droplet collision
simulations."

> page 8, line 176: it seems that this is the first mention of
> dimensionality of presented simulations, better to state it when
> introducing particle attributes (i.e. xi, vi)

We have now moved it down to the last paragraph of section 2.b.

> page 9, line 180: "twice the mass and radius" - please rephrase

We have now corrected it as "twice the mass, so that the radius is".

> page 9, line 190-194: perhaps worth referencing here the discussion
> on common/rare super-particle sub- population sampling from DeVille et
> al. 2019, section 6.1 therein

This jumps in our study are due to the superdroplet collision scheme.
To our knowledge, it is not related to 
rare sub-population sampling of particles in DeVille et al. (2019).

> page 11: line 242-243: change square brackets into normal parenthesis

Changed.

> page 13, line 281: this is the first mention of mean-field theory
> in the paper, please elaborate, clarify, reference works which provide
> more details

What we meant by MFT is that the actual collision times are just
replaced by the mean collision times that are given by t_k=lambda_k^{-1}.
To clarify this better, we have now written: "By comparison, if fluctuations
are ignored, the collision times that are given by t_k=lambda_k^{-1}.
This is what we refer to as mean-field theory (MFT)."

> page 13, fig 2: use logarithmic sampling for the curves so that in
> the left part of the plot the curves are smooth

We have now corrected this.

> page 13, fig 3: ditto

The unsmooth appearance was mainly due to the inclusion of the time T_k=0,
which we have now removed. We recall that this case is slightly different
from that of Figure 2 in that we talk here about discrete times.

> page 14, line 290: "approach I" mentioned before being defined

We agree with the referee that the word "approach I" has now been used
too early, This was also noticed by referee I, and we have now removed
the reference to approach I in the first location and in the captions.

> page 14, lines 291-293: the mention of Pencil Code here (in the
> "Relaxing the power law approximation" section) seems misplaced

We have now revised it as
"We refer to appendix A1 for details of performing this many realizations."

> page 15, line 319: it is unclear for me what does it mean for a
> distribution to be "somewhat enhanced"

We have now rephrased it to "..., indicating that the distribution
broadens". 

> page 16, line 322: "is" -> "are"

Corrected.

> page 16, line 328: "Here and below" - unclear

We have now moved this paragraph describing why plotting P(T/<T>) to
the caption of Fig.4, when P(T/<T>) appears for the first time.

> page 16, line 334: this sentence would be best moved to the first
> paragraphs of the paper with the aim of clarifying the dimensionality
> aspect.

This entire paragraph fits better here because it is an extension of LDM.
The dimensionality aspect is now introduced in section 2.b early in the paper.

> page 17, line 342: "at the end of this paper", point precisely to a section

We have now specified it as "at section 4.d".

> page 18, line 363: "collision partner" not introduced earlier,
> if embracing such notion, worth to use it when describing the algorithm
> in the beginning of the paper

We have now introduced the "collision partner" just above Ea.3, when the
superdroplet algorithm is first introduced.

> page 19, line 395: "fat" --> "thick"

Changed.

> page 20, line 96: please rephrase "In its simplest form" being
more precise (same on page 26, line 231)

We have now rephrased it as "The LDM assumes that the ..." and "The LDM ...".

> page 21, line 402 and 427: there is no green line, green points?

Corrected.

> page 22, line 437: to reduce computational cost?

Rephrased.

> page 23, line 452: unclear if "section 1" here of in D&P

we have now rephrased as "as we discussed in section 1."

> page 23, line 453: please be more specific than "at late times"

We have now made it more specific as the following, 
"... at the last few steps
of a lucky droplet growing to 50 um (see Figure 9) ..."

> page 24, line 469: 1/255 ~ 0.004

Corrected.

> page 25, line 496: there seem to be no "dotted" line in the plot

We have now corrected it as "thick black line". 

> page 27, line 521: "does not contain mean-field elements" is unclear

We have now elaborated on it as
"... is able to represent fluctuations during collisions and
does not contain mean-field elements".

> page 27, line 529: "appear to be vastly exaggerated" - be more specific

We have now removed this statement.

> page 29, line 564: rephrase "authors point out", "them having
> chosen" with non-personal wording

Rephrased.

> page 29, line 571: avoid "believe" wording

We've now rephrased it as "does not hold in this
investigation".

> page 31, line 610: move code location from Acknowledgements to the
> "Data availability statement"

Moved.

> page 31, line 612: mention that the archive also contains
> "plotting/analysis scripts and that the data is stored in a
> proprietary "sav" format

Added.

> page 31, line 613: missing "doi.org" in the url

Added.

> page 32, line 645: what is "usual LDM"?

We have now removed "usual".

> page 39, line 799: "Journal of Atmospheric Sciences" --> missing "the"

Added.

> page 39, line 806: "Physics Review Letter" --> "Physical Review Letters"

Corrected.